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Journal of Inequalities and Applications

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  • Research Article
  • Open Access

An impulsive nonlinear singular version of the Gronwall-Bihari inequality

Journal of Inequalities and Applications20062006:84561

https://doi.org/10.1155/JIA/2006/84561

©  Tatar 2006

Received: 11 August 2005

Accepted: 20 October 2005

Published: 7 June 2006

Abstract

We find bounds for a Gronwall-Bihari type inequality for piecewise continuous functions. Unlike works in the prior literature, here we consider inequalities involving singular kernels in addition to functions with delays.

Keywords

Continuous FunctionType InequalityPrior LiteraturePiecewise Continuous FunctionSingular Kernel

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Authors’ Affiliations

(1)
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

References

  1. Bainov DD, Hristova SG: Impulsive integral inequalities with a deviation of the argument. Mathematische Nachrichten 1995, 171: 19–27.MathSciNetView ArticleMATHGoogle Scholar
  2. Bainov DD, Simeonov PS: Systems with Impulse Effect: Theory and Applications. Ellis Horwood, Chichister; 1989.MATHGoogle Scholar
  3. Bainov DD, Simeonov PS: Integral Inequalities and Applications, Mathematics and Its Applications. Volume 57. Kluwer Academic, Dordrecht; 1992:xii+245.View ArticleMATHGoogle Scholar
  4. Butler G, Rogers T: A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations. Journal of Mathematical Analysis and Applications 1971,33(1):77–81. 10.1016/0022-247X(71)90183-1MathSciNetView ArticleMATHGoogle Scholar
  5. Hristova SG: Nonlinear delay integral inequalities for piecewise continuous functions and applications. Journal of Inequalities in Pure and Applied Mathematics 2004,5(4):1–14. article 88 article 88MathSciNetMATHGoogle Scholar
  6. Kirane M, Tatar N-E: Global existence and stability of some semilinear problems. Archivum Mathematicum (Brno) 2000,36(1):33–44.MathSciNetMATHGoogle Scholar
  7. Kirane M, Tatar N-E: Convergence rates for a reaction-diffusion system. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2001,20(2):347–357.MathSciNetView ArticleMATHGoogle Scholar
  8. Krylov NN, Bogolyubov NN: Introduction to Nonlinear Mechanics. Izd. Acad. Sci. Ukr. SSR, Kiev; 1937.Google Scholar
  9. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, New Jersey; 1989:xii+273.View ArticleGoogle Scholar
  10. Mazouzi S, Tatar N-E: Global existence for some integro-differential equations with delay subject to non-local conditions. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2002,21(1):249–256.MathSciNetView ArticleMATHGoogle Scholar
  11. Medved M: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. Journal of Mathematical Analysis and Applications 1997,214(2):349–366. 10.1006/jmaa.1997.5532MathSciNetView ArticleMATHGoogle Scholar
  12. Medved M: Singular integral inequalities and stability of semilinear parabolic equations. Archivum Mathematicum (Brno) 1998,34(1):183–190.MathSciNetMATHGoogle Scholar
  13. Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, California; 1998:x+611.MATHGoogle Scholar
  14. Samoilenko AM, Perestyuk NA: Stability of solutions of differential equations with impulse effect. Differential Equations 1977,13(11):1981–1992.Google Scholar
  15. Samoilenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, New Jersey; 1995:x+462.Google Scholar
  16. Tatar N-E: Exponential decay for a semilinear problem with memory. Arab Journal of Mathematical Sciences 2001,7(1):29–45.MathSciNetMATHGoogle Scholar
  17. Zhang W, Agarwal RP, Akin-Bohner E: On well-posedness of impulsive problems for nonlinear parabolic equations. Nonlinear Studies 2002,9(2):145–153.MathSciNetMATHGoogle Scholar

Copyright

© Tatar 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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